Spectral Geometry and the Kaehler Condition for Hermitian Manifolds with Boundary

Let (M,g,J) be a compact Hermitian manifold with a smooth boundary. Let ${\mathrm{\Delta }}_p$ and $D_p$ be the realizations of the real and complex Laplacians on p forms with either Dirichlet or Neumann boundary conditions. We generalize previous results in the closed setting to show that (M,g,J) is Kaehler if and only if $Spec({\mathrm{\Delta }}_p)=Spec(2D_p)$ for p=0,1. We also give a characterization of manifolds with constant sectional curvature or constant Ricci tensor (in the real setting) and manifolds of constant holomorphic sectional curvature (in the complex setting) in terms of spectral geometry.